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In this article I am going discuss how to think about centripetal motion and weightlessness and talk about some common misconceptions that cause confusion. This article is probably of most use to people who have come across centripetal motion before. It introduces some concepts that are not covered in the A-level physics curriculum, but which are useful for thinking about what's going on.

Centripetal MotionFirst of all, let's be clear about what centripetal motion is. Centripetal motion is when an object moves in a circle without changing its speed. If the constant speed of the moving object is known — let's call it v — and the radius of the circle that it is moving in is also known — let's call it r — then it can be shown that the object's acceleration has magnitude a = v2 / r, is constant, and always points towards the centre of the circle, perpendicular to the motion of the object around the circumference.

Whilst I won't derive it here (it takes a bit of trigonometry and calculus) this is a mathematical result that always holds. It's important to realise that whenever you have centripetal motion then you must necessarily have this perpendicular acceleration; and equally that whenever you have an acceleration that doesn't change its magnitude and always stays perpendicular to motion then you will necessarily have centripetal motion and the size of the circle that the object will move in will be determined by the speed v and magnitude of the acceleration a — rearranging the formula above gives the radius r as v2 / a. Conversely, if the acceleration is either not always perpendicular to motion or not always of constant magnitude then we necessarily do not have centripetal motion — the motion must either not be circular or not at a constant speed or both.

The acceleration of the object a is of course caused by a resultant force on the object F and the relationship between them is the familiar F = ma, where m is the mass of the object. We call this force the centripetal force. Since the centripetal force is always perpendicular to the direction moved, constantly adjusting, the work done by the force (force times the distance moved in direction of the force) always remains zero; this is a way to see that a centripetal force does not increase or decrease the object's kinetic energy and that the object therefore keeps a constant speed as the force pulls it round in a circle.

WeightlessnessNow, let's also be clear about what we mean by weightlessness. By weightlessness we mean just the floating that you are probably imagining. If you are weightless then you are floating with respect to your surroundings and the different part of yourself are all floating with respect to each other — you have the odd sensation of your insides floating inside you. You experience this weightless feeling if you are in a car that ramps off the top of a hill, if you are floating in deep space, or if the ground beneath you suddenly gives way and you find yourself falling.

What this should make clear is that, confusingly, by weightlessness we do not mean the absence of the weight force (mass times acceleration due to gravity). Clearly the weight force from gravity is always acting down here on Earth, yet two of the above examples of feeling weightless occur down here on Earth. To understand this it will be useful to consider the classic example of standing in a lift. When the lift is stationary the weight force acting on you from gravity is balanced by the contact force of the floor pushing you up. These contact forces are also present throughout your body — your legs are holding up the rest of your body, your neck is holding up your head, and so on. But if the cable breaks and the lift and you inside of it start falling under gravity then these contact forces go away. The lift and every part of your body are all accelerated by the same acceleration due to gravity and since they all share in the falling motion none of the parts need to be held up by contact forces. In a freefalling lift you experience weightlessness.

What's more, if you cannot see outside of the lift, and have no memories of how you got there, it is impossible to tell if you and the lift are in freefall or if you and the lift are floating in deep space with no gravity at all. In fact, weightless scenes for movies are often filmed inside of a freefalling aeroplane.

In a less extreme situation, when you are in a lift and it starts accelerating downwards to get to a lower floor you feel somewhere between normal and weightless; that is, when the lift starts moving downwards you feel slightly lighter. This only lasts while the lift is accelerating downwards; once it reaches its top speed then you feel normal again. When the lift starts accelerating upwards, to slow its downwards motion and come to a stop, you feel slightly heavier, until the lift stops and the upwards acceleration stops.

Circular OrbitsWith that clear, we can now think about why everyone is weightless on the International Space Station (ISS). It is not because there is no gravity! The station is only about 400 kilometres up, compared to the Earth's radius of about 6370 kilometres, and the acceleration due to gravity — g — is about 89% as strong up there as it is down here at the surface. The reason that they are weightless is because the station and the people inside are in freefall. Just like in the falling lift example, the only force acting on them is the force due to gravity. So, they are in freefall, but they don't actually lose any height. Let's try to understand this.

The space station is moving in a circle around the Earth with a constant speed. That's centripetal motion. So what is the centripetal force? It's important to understand that the "centripetal force" is not a type of force like "the weight" or "the normal contact force / normal reaction" or "the tension" or "the friction". Rather, the label "centripetal" just tells you what the force is doing, not what causes it. Any resultant force can be a centripetal force — it just means that it is perpendicular to the object's motion and of constant magnitude and therefore causes centripetal motion. So for the space station the centripetal force keeping it in a circle is exactly the gravitational force towards the Earth. They are one and the same.

Since we know that the space station is 400 km up we can find the acceleration caused by this gravitational force to be g = 8.7 m/s2. And the radius of its orbit is R = 6370 km + 400 km = 6770 km. And since we know that we have centripetal motion we can use the equation g = v2 / R to find the speed v — rearranging we find that it's the square root of the radius times the acceleration, v = √(gR). This works out at 7.4 kilometres per second, which is how fast the space station is travelling around the Earth. note that since the orbit is circular and the height remains constant, the gravitational force towards the centre of the Earth is always perpendicular to motion and always of the same strength, as required for centripetal motion.

We can say that the space station is not falling down, but rather is falling around the Earth, having enough sideways speed that gravity just keeps it in its circular orbit. And since the station and the astronauts all share in this centripetal acceleration due to gravity there are no contact forces required and everyone is weightless.

Non-Circular OrbitsSo what would happen if the speed of the space station up there were something different, say something smaller than 7.4 km/s? Well, since g and R would be unchanged the equation g = v2/R would no longer be true, so the motion would not be centripetal. The station would no longer move in a circular orbit, but would start to lose height. Since it would no longer be moving in a circle around the Earth and the acceleration towards the centre of the Earth would no longer be constant or perpendicular to its motion it would start to speed up as it got closer to the Earth. I shall not derive it here, but it turns out that once it got to the other side of the Earth the station would actually be moving fast enough to start gaining height (and losing speed) again and would eventually come back to its original position, tracing the shape of an ellipse — a motion that it would continually repeat.

It turns out that when the conditions for centripetal motion are not exactly satisfied then orbits are elliptical in shape, or the object may be moving too fast to be held in any orbit at all. If the speed were larger than 7.4 km/s but still slow enough to be in held in orbit then the station would gain height at first and get slower, starting to descend again only once it got to the other side of the Earth. If the speed around the Earth were zero then the station would not orbit at all and would simply fall to Earth. This corresponds to an infinitely flat ellipse, which is just a straight line. With a small non-zero orbital speed the station would move around the Earth a little while falling, but still might not produce a fat enough ellipse to avoid encountering the Earth's atmosphere. On the other hand, if the station were moving really fast the Earth's gravity would not be enough to pull it round very much before it got a long way away and in this case the station might not be bound in an orbit at all — and if it was then it would take a very long time to get to the other side of the Earth since it would get very slow and far away.

In our Solar System the planets are all on nearly circular (centripetal) orbits around the Sun, whereas comets are on highly elliptical orbits — they are quite fast moving when they come in to swing around the Sun, but spend the vast majority of their time moving very slowly a long way away.

Centrifugal ForcesSo, astronauts, whether they are in orbit or whether they are in deep space with no gravity at all, have to contend with weightlessness, which, as it turns out, is quite damaging to humans. There is, however, one very plausible idea for artificial gravity — the spinning space station. Imagine that there is a station in deep space that is shaped like a hollow cylinder or a giant tin can with you standing on the inside of the curved surface; we will explain why you are standing and not floating presently.

If the whole cylinder is spinning at a constant rate and you remain at your point on the curved surface then you must be moving in a circle with a constant speed and that's centripetal motion. So what is the centripetal force in this case? The answer is the contact force from the curved surface, constantly pushing you to keep you inside the station and pushing you round in a circle. If not for this contact force the speed that you had from the spinning would just carry you in a straight line. Since you are constantly experiencing a contact force from the surface, it is just like standing on the ground down here on Earth. Here we have an example where you don't feel weightless, but there is no actual gravity at all.

The size of this contact force would depend on how fast the station was spinning — v — and how large its radius was — r — but if v2 / r were equal to the acceleration due to gravity down here on Earth — g — then the contact force on you from the inside surface of the station would be equal to your mass times g, just like the contact force from the solid ground is down here to balance your weight.

It's interesting to think about how things would look to you inside the spinning station if you couldn't see out and again didn't have any memories of how you got there. From your point of view, relative to yourself, you are not moving, but you are experiencing a contact force from the floor beneath you. You would therefore conclude that there must be a weight force acting "downwards" (away from the centre of the station) to be balanced by this contact force. This force is called the centrifugal force and whether or not it exists quite literally depends on your point of view.

The centrifugal force is known as a "fictitious" force. In our original analysis this force didn't appear; there was just a contact force acting as a centripetal force. The centrifugal force only shows up when we reformulate physics from the point of view of someone who is moving in a circle. We call that a "rotating reference frame" and in order to make the laws of physics consistent we have to introduce fictitious forces when we consider things from the rotating reference frame's point of view. So, from the point of view of an outside observer the contact force is acting as a centripetal force and moving you round in a circle as the station rotates; but from your point of view you are not moving at all and the contact force is holding you up against a centrifugal force (of the same size as the centripetal force) that acts just like gravity, trying to push you out through the station's floor (and down/out into space).

If you're wondering about the centrifugal force on the ISS and its astronauts because of their orbiting the Earth — it does exist. It is exactly equal and opposite to the gravitational force that is acting as the centripetal force in this case. So from the point of view of someone on board they are not moving (rather the Earth is turning beneath them) and they are weightless because the gravity force is exactly balanced by the centrifugal force.

The Earth's RotationSo, what about the daily rotation of the Earth itself? Shouldn't that cause a centrifugal force that tries to throw us off of the surface and into space? The answer is "yes", but it happens that this force is quite small.

Let's imagine that you are standing on the equator, to make things easier. Since the Earth rotates once every 24 hours, you must be moving in a circle the size of the equator every 24 hours. The circumference of the equator is about 40,000 km, so your speed must be 40,000 km / 24 hours = 1670 km/hour. That sounds pretty fast, but let's see what centripetal acceleration this implies. We get (1670 km/hour)2 / (6370 km) = (1,670,000 m / 3600 s)2 / (6,370,000 m) = 0.034 m/s2. When we compare this to the acceleration due to gravity of about 9.8 m/s2 this is actually pretty small.

But, small though it is, there must be a resultant force on us that causes this centripetal acceleration. So what is it? It is actually just a small difference between the gravity force and the contact force from the ground that causes this small resultant force. And from your point of view you are not moving at all and the gravity pulling you down is balanced by a small centrifugal force plus the contact force from the ground making up the rest.

It turns out that the Earth would need to rotate about once every 84 minutes (we would have 84 minute long days) in order for the necessary centripetal acceleration (to keep you at the same place on the Earth's surface as it rotated) to be equal to the acceleration due to gravity. If the Earth were rotating this quickly then the entirety of the gravitation force would be needed to make up the larger centripetal force and there would be no contact force from the ground. You would be floating at the Earth's surface. From your point of view you would not be moving and the gravitational force would be exactly balanced by the centrifugal force, so you would feel weightless and would need no contact force from the ground to hold you up. This is exactly the same as being in orbit at the Earth's surface, with the same part of the Earth always beneath you. There wouldn't even be any air resistance as long as the air was also moving at the same speed as the surface.

If the speed of rotation become even faster then the acceleration due to gravity would be insufficient to make up the required centripetal acceleration and you would not be kept in your orbit. You would be thrown into space by the fast rotation of the ground. From your point of view, you would simply float up into space as the outwards centrifugal force would be larger than your weight.

SummaryHopefully the examples discussed here will help you to avoid the usual traps when thinking about centripetal motion and weightlessness. Centripetal motion with speed v and radius r implies and constant inwards acceleration of magnitude a = v2 / r; and a constant acceleration of magnitude a perpendicular to an object's motion with speed v implies centripetal motion with a radius r = v2 / a. Centripetal motion around a circle with radius r with a constant inwards acceleration of magnitude a implies a speed of v = √(ar) and if the speed is something different then the motion will not be centripetal.

To tell whether something is weightless it is sufficient to examine whether any contact forces are needed to hold it up or not, remembering that the centripetal force is not a separate kind of force, but merely the name given to a resultant force that produces centripetal motion. It is not necessary to consider things from the point of view of an object that is moving in a circle — the rotating reference frame — but if you do then you must remember that from that object's point of view, relative to itself, the object is not moving and you must remember to include the fictitious centrifugal force in order for the laws of physics to be consistent.